If I had nowhere else to put a 9' table and had no way to somehow widen that 13' room, here's what I'd do.
I'd stick with the 9-footer rather than getting a smaller table, and I'd place it off-center in the room. I'd place one of the long sides of the table so the edge of the cushion (edge of the 50" wide playing surface) is 66" from the wall. That will put the opposite cushion edge just 40" from the other side wall. 66" + 50" + 40" = 156" = 13'.
That will give you full-cue/full-backswing access to three sides of the table plus some shots on the fourth side. For shots where a full cue could not be used on that fourth side, you could use a shorter cue.
Obviously, this is not ideal, and may even sound kind of crazy. But I think I'd find it plenty acceptable for the way I use a home table, which is for practice more than for playing games with other people. With the full use of three sides, you would have no limitation on the types of shots you could practice.
You're lucky to have the 9' GC IV. I'd want to use that to better my game regardless of any room limitations. Good luck, whatever you do.